论文标题
Graham-Knuth- -Patashnik复发：对称性和持续分数
The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions
论文作者
论文摘要
我们研究由Graham-knuth--patashnik复发$ t（n，k）=（αn +βk +γ）\，t（n-1，k） +（α'n +β'k +γ'）\，t（n-1，k +γ'）\，t（n-1，k-1）$具有初始条件$ t（0，K）$ t（0，K） =（α，β，γ，α'，β'，γ'）$。我们表明，在48个元素离散组同构至$ s_3 \ times d_4 $下，阵列$ t（\mathbfμ）$的家族是不变的。我们的主要结果是确定所有参数集$ \MATHBFμ\ in \ Mathbb {C}^6 $，其中普通生成函数$ f（x，x，x，t）= \ sum_ {n，k = 0}^\ ind}^\ infty t（n，n，k）\，x^k t^n $继续由stieltjes-type $ the $ the $ the $ the $ $ x $。我们还展示了一些特殊情况，其中$ f（x，t）$由thron型或jacobi-type持续分数为$ t $，其系数为$ x $。
We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (αn + βk + γ)\, T(n-1,k)+(α' n + β' k + γ') \, T(n-1,k-1)$ with initial condition $T(0,k) = δ_{k0}$ and parameters $\mathbfμ = (α,β,γ, α',β',γ')$. We show that the family of arrays $T(\mathbfμ)$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbfμ \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coefficients that are polynomials in $x$. We also exhibit some special cases in which $f(x,t)$ is given by a Thron-type or Jacobi-type continued fraction in $t$ with coefficients that are polynomials in $x$.
